Lie bracket from local group commutators ($[X,Y]={\partial }_s{\partial }_t\log (e^{sX}{e}^{tY}{e}^{-sX}{e}^{-tY}){|}_{s=t=0}$)

In a local exponential chart, the mixed second differential of the group commutator defines a bilinear map on the tangent space at the identity. Inversion after swapping the two group elements proves antisymmetry. Differentiating conjugation gives $[X,Y]=\mathrm{ad}(X)Y$; applying naturality of the Lie bracket to the Adjoint representation of a Lie group then proves the Jacobi identity. For a matrix group this recovers $XY-YX$ by expanding through the mixed term.